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The nilpotency of some groups with all subgroups subnormal. (English) Zbl 0921.20033
Let \(G\) be a group with all subgroups subnormal. If \(\langle x\rangle^G\) is finitely generated for each \(x\in G\) then \(G\) is nilpotent [H. Smith, Glasg. Math. J. 36, No. 1, 33-36 (1994; Zbl 0803.20019)]. The article under review is dedicated to the following question: what other conditions on \(\langle x\rangle^G\) imply the nilpotency of \(G\)? Let \(G\) be a group, \(H\) a normal subgroup of \(G\). Then \(H\) is said to be \(G\)-minimax, if \(H\) has a finite series of \(G\)-invariant subgroups every factor of which is abelian and satisfies Max-\(G\) or Min-\(G\).
The main results of this paper are the following theorems. Theorem A. Let \(G\) be a group with all subgroups subnormal. If \(\langle x\rangle^G\) is \(G\)-minimax for all \(x\in G\) then (1) \(G\) is nilpotent; (2) \(\langle x\rangle^G\) is minimax for all \(x\in G\). Theorem B. Let \(G\) be a group with all subgroups subnormal. If \(\langle x\rangle^G\in{\mathbf S}_1\) then \(G\) is nilpotent. Corollary. Let \(G\) be a group with all subgroups subnormal. If \(G\) has minimax conjugacy classes then \(G\) is nilpotent.
MSC:
20E15 Chains and lattices of subgroups, subnormal subgroups
20F18 Nilpotent groups
20F05 Generators, relations, and presentations of groups
20F19 Generalizations of solvable and nilpotent groups
20F22 Other classes of groups defined by subgroup chains
Citations:
Zbl 0803.20019
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